bibkey: ganardi2022hierarchy authors: Ray Ganardi; Marek Miller; Tomasz Paterek; Marek Żukowski year: 2022 title: “Hierarchy of correlation quantifiers comparable to negativity” doi: 10.22331/q-2022-02-16-654 url: https://arxiv.org/abs/2111.11887v2 claim: “For a bipartite state rho with partial transpose rho^{T_B} on the second system, the negativity is N(rho) = (||rho^{T_B}||_1 - 1)/2 and the partial transpose distance is d_T(rho, sigma) = ||rho^{T_B} - sigma^{T_B}||_1 / 2. Conjecture 1: for every density matrix rho, the infimum of d_T(rho, sigma) over the PPT states sigma equals N(rho).” strata_touched:

• D5/S3/Quantum/Entanglement/NegativityPartialTransposeDistanceRefutation license: citation-only triage: anchor

Hierarchy of correlation quantifiers comparable to negativity

Ray Ganardi, Marek Miller, Tomasz Paterek, Marek Żukowski, Quantum 6, 654 (2022); arXiv:2111.11887v2 (quant-ph). Quotations are from the arXiv v2 source, with its macros kept as printed (\pt{\rho} is ${\rho }^{T_B}$, \norm{A} is $\Vert A{\Vert }_1$, \dm is the set of density matrices on ${\mathcal{H}}_A\otimes {\mathcal{H}}_B$).

The partial transpose and the trace norm:

Given a bipartite system \hs_A \otimes \hs_B, we denote partial transpose on the computational basis of \hs_B as $T_B$.

The trace norm of a matrix $A$ is defined as \norm{A} = \Tr{\sqrt{A^\dagger A}}.

The PPT states and the negativity:

A state $\rho$ is PPT if \pt{\rho} \geq 0, and we denote the collection of such states as \PPT.

The negativity of a state $\rho$ is defined as N(\rho) = \half \left(\norm{\pt{\rho}} - 1\right).

The partial transpose distance (Definition):

Let \rho, \sigma \in \dm be two density matrices. The partial transpose distance $d_T(\rho ,\sigma )$ is defined as \begin{align} d_T(\rho, \sigma) &= \half \norm{\pt{\rho} - \pt{\sigma}}. \end{align}

Conjecture 1:

Let $\rho$ be a density matrix. Then \begin{align*} \inf_{\sigma \in \PPT} d_T(\rho, \sigma) &= N(\rho). \end{align*}

The paper proves the equality for states of positive binegativity, $|{\rho }^{T_B}{|}^{T_B}\ge 0$, and reports that $10^6$ random mixed states of two qudits for each $d=2,\dots ,6$ show no counterexample.

The counterexample state of the settlement belongs to the punch-card family of L. Lami, F. A. Mele, B. Regula, arXiv:2405.09613v2 (Phys. Rev. Lett. 134, 090202 (2025)), Eq. (15), where the family is used for the additivity of the κ-entanglement; the separation of the PPT distance from the negativity is not stated there.

The encoding takes states on ${\mathbb{C}}^d\otimes {\mathbb{C}}^d$, indexes matrices by Fin d × Fin d, transposes the second factor, and uses the trace norm $\mathrm{Re}\mathrm{Tr}\sqrt{A^{†}A}$; the negativity and the distance are written in the claim exactly as defined above.

Verified locator

• DOI: https://doi.org/10.22331/q-2022-02-16-654 (Quantum 6, 654 (2022)).
• URL: https://arxiv.org/abs/2111.11887v2 (v2, 2022-02-09, the latest version; source main.tex, md5 df386f803c4233dab0a6307e9c4dc578): the partial transpose (l. 153), the trace norm (l. 155), the PPT states (l. 159), the negativity (l. 163), the partial transpose distance (l. 172–177), Conjecture 1 (l. 268–274) and the numerical evidence (l. 316–320).